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We analyze a recently proposed model [I. Averbukh et al., Development, 141 (2014), pp. 2150-2156 for the regulation of growth and patterning in developing tissues by diffusing morphogens. We show that solutions of the underlying coupled systems of nonlinear PDEs exist, are unique, and are stable in a suitable sense. The key tool in the analysis is the transformation of the underlying system to a porous medium equation. Numerical experiments illustrating the model are also presented.

In this paper the existence of a positive measurable solution of the Hammerstein equation of the first kind with a singular nonlinear term at the origin is presented.

The long time behavior of a model for the regulation of growth and patterning in developing tissues by diffusing morphogens is analyzed. Such model is expressed in terms of a system of nonlinear PDEs. The key tool in the analysis is the transformation of such system to an equation with singular diffusion. Keywords Singular diffusion Á Neumann boundary conditions Á Long time behavior Á Morphogen evolution Mathematics Subject Classification 35K55 Á 35K65 Á 35Q92 Á 34B15 Á 35B40 • How can a diffusing morphogen concentration scale with growing tissue? This article is part of the section ''Applications of PDEs'' edited by Hyeonbae Kang.